ALBERT

Description: The stability of the viscometric motion of a viscoelastic fluid held between rotating parallel disks with large radii to small-amplitude perturbations is studied for the Oldroyd-B constitutive model. The disturbances are assumed to be radially localized and are expressed in Fourier form so that a separable eigenvalue problem results; these disturbances describe either axisymmetric or spiral vortices, depending on whether the most dangerous disturbance has zero or non-zero azimuthal wavenumber, respectively. The critical value of the dimensionless radius R* for the onset of the instability is computed as a function of the Deborah number De, a dimensionless time constant of the fluid, the azimuthal and radial wavenumbers, and the ratio of the viscosities of the solvent to the polymer solution. Calculations meant to match the experiments of McKinley et al. (1991) for a Boger fluid show that the most dangerous instabilities are spiral vortices with positive and negative angle that start at the same critical radius and travel outward and inward toward the centre of the disk; the axisymmetric mode also becomes unstable at only slightly greater values of R*, or De for fixed R*. The predicted dependence of the value of De for a fixed R* on the gap between the disks agrees quantitatively with the measurements of McKinley et al., when the longest relaxation time for the fluid at the shear rate corresponding to the maximum value of R* is used to define the time constant in the Oldroyd-B model. © 1993, Cambridge University Press. All rights reserved.

Description: Experimental observations and linear stability calculations are presented for the stability of torsional flows of viscoelastic fluids between two parallel coaxial disks, one of which is held stationary while the other is rotated at a constant angular velocity. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the purely circumferential, viscometric base flow becomes unstable with respect to a nonaxisymmetric, time-dependent motion consisting of spiral vortices which travel radially outwards across the disks. Video-imaging measurements in two highly elastic polyisobutylene solutions are used to determine the radial wavelength, wavespeed and azimuthal structure of the spiral disturbance. The spatial characteristics of this purely elastic instability scale with the rotation rate and axial separation between the disks; however, the observed spiral structure of the secondary motion is a sensitive function of the fluid rheology and the aspect ratio of the finite disks. Very near the centre of the disk the flow remains stable at all rotation rates, and the unsteady secondary motion is only observed in an annular region beyond a critical radius, denoted R*1. The spiral vortices initially increase in intensity as they propagate radially outwards across the disk; however, at larger radii they are damped and the spiral structure disappears beyond a second critical radius, R*2. This restabilization of the base viscometric flow is described quantitatively by considering a viscoelastic constitutive equation that captures the nonlinear rheology of the polymeric test fluids in steady shearing flows. A radially localized, linear stability analysis of torsional motions between infinite parallel coaxial disks for this model predicts an instability to non-axisymmetric disturbances for a finite range of radii, which depends on the Deborah number and on the rheological parameters in the model. The most dangerous instability mode varies with the Deborah number; however, at low rotation rates the steady viscometric flow is stable to all localized disturbances, at any radial position. Experimental values for the wavespeed, wavelength and azimuthal structure of this flow instability are described well by the analysis; however, the critical radii calculated for growth of infinitesimal disturbances are smaller than the values obtained from experimental observations of secondary motions. Calculation of the time rate of change in the additional viscous energy created or dissipated by the disturbance shows that the mechanism of instability for both axisymmetric and non-axisymmetric perturbations is the same, and arises from a coupling between the kinematics of the steady curvilinear base flow and the polymeric stresses in the disturbance flow. For finitely extensible dumb-bells, the magnitude of this coupling is reduced and an additional dissipative contribution to the mechanical energy balance arises, so that the disturbance is damped at large radial positions where the mean shear rate is large. Hysteresis experiments demonstrate that the instability is subcritical in the rotation rate, and, at long times, the initially well-defined spiral flow develops into a more complex three-dimensional aperiodic motion. Experimental observations indicate that this nonlinear evolution proceeds via a rapid splitting of the spiral vortices into vortices of approximately half the initial radial wavelength, and ultimately results in a state consisting of both inwardly and outwardly travelling spiral vortices with a range of radial wavenumbers. © 1994, Cambridge University Press. All rights reserved.

Description: The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio B, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 〈 b〈 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions. © 1991, Cambridge University Press. All rights reserved.

Description: A set of microstructural variables is selected to characterize the behavior of snow. Corresponding mathematical relations from quantitative stereology theory are presented along with relations and techniques required for numerical evaluation. An experimental investigation is carried out to determine changes in these variables for snow subjected to large compressive deformations. The micro-structural variables studied included coordination number, grain-size, bond radius, neck length, pore-size, free surface area and grains/unit volume. Measurements at several stages of deformation are used to evaluate the changes in the microstructure as functions of deformation. Microstructure measurements of six snow samples subjected to confined compression tests are presented for pre-compressed and compressed states, corresponding to final stresses of 0.387, 0.77 and 1.55 MPa. Grain-size and bond radius were found to go through finite changes during compression, although the variation of bond radius was more complicated in nature. The coordination number and number of bonds/unit volume were found to go through large changes during compression, while specific free surface area was found to increase by 100% due to grain- and bond-fracture processes. No discernible patterns of change in neck length could be found in the experiments. A close relationship between some of the microstructural variables and the stress response of the material was observed. These results serve to contribute to the presently available data and understanding of the microstructural behavior of snow.

Description: A set of microstructural variables is selected to characterize the behavior of snow. Corresponding mathematical relations from quantitative stereology theory are presented along with relations and techniques required for numerical evaluation. An experimental investigation is carried out to determine changes in these variables for snow subjected to large compressive deformations. The micro-structural variables studied included coordination number, grain-size, bond radius, neck length, pore-size, free surface area and grains/unit volume. Measurements at several stages of deformation are used to evaluate the changes in the microstructure as functions of deformation. Microstructure measurements of six snow samples subjected to confined compression tests are presented for pre-compressed and compressed states, corresponding to final stresses of 0.387, 0.77 and 1.55 MPa. Grain-size and bond radius were found to go through finite changes during compression, although the variation of bond radius was more complicated in nature. The coordination number and number of bonds/unit volume were found to go through large changes during compression, while specific free surface area was found to increase by 100% due to grain- and bond-fracture processes. No discernible patterns of change in neck length could be found in the experiments. A close relationship between some of the microstructural variables and the stress response of the material was observed. These results serve to contribute to the presently available data and understanding of the microstructural behavior of snow.

Description: A continuum theory of mixtures is applied to model snow as a mixture of an elastic solid and an elastic fluid. Three wave types, two dilational and one rotational, are shown to exist. Numerical evaluation shows velocity and attenuation increasing with frequency for all three waves. Wave velocity increases with increasing density while attenuation decreases with increasing density for all three waves. The first dilational wave is associated with the pore fluid, has a slow wave speed and is highly attenuated. This wave exhibits diffusive behavior at low frequencies and nondispersive behavior at high frequencies. The second dilation wave is associated with the solid ice material. It is the fastest of the three wave types and does not appreciably attenuate. Nondispersive wave behavior characterizes this wave at low and high frequencies. The rotational wave occurs only in the solid, is the least attenuated of all three waves, and propagates at velocities greater than that of the first, but less than that of the second, dilational wave. The rotational wave exhibits nondispersive behavior at low and high frequencies. Wave velocities and attenuation show behavior that is in agreement with existing experimental data.

Description: A continuum theory of mixtures is applied to model snow as a mixture of an elastic solid and an elastic fluid. Three wave types, two dilational and one rotational, are shown to exist. Numerical evaluation shows velocity and attenuation increasing with frequency for all three waves. Wave velocity increases with increasing density while attenuation decreases with increasing density for all three waves. The first dilational wave is associated with the pore fluid, has a slow wave speed and is highly attenuated. This wave exhibits diffusive behavior at low frequencies and nondispersive behavior at high frequencies. The second dilation wave is associated with the solid ice material. It is the fastest of the three wave types and does not appreciably attenuate. Nondispersive wave behavior characterizes this wave at low and high frequencies. The rotational wave occurs only in the solid, is the least attenuated of all three waves, and propagates at velocities greater than that of the first, but less than that of the second, dilational wave. The rotational wave exhibits nondispersive behavior at low and high frequencies. Wave velocities and attenuation show behavior that is in agreement with existing experimental data.

Description: A constitutive theory of snow is developed to describe the mechanical properties of snow in terms of the properties of the ice grains and the necks that interconnect them. The principle of virtual work is used to calculate the stresses in the particles and necks. A number of different deformation mechanisms are investigated and, depending upon the deformation mechanism which is dominant for given load conditions, different equations are used to calculate the strains in the grains and necks. These strains around a representative ice grain are then averaged and scaled to obtain the global strains in the snow. The theory is then compared with experimental data to determine if the mechanical properties of snow can be adequately represented. Results show that the constitutive theory does work, but that it is cumbersome to implement, and that for practical use substantial computational capability is needed.

Description: A constitutive theory of snow is developed to describe the mechanical properties of snow in terms of the properties of the ice grains and the necks that interconnect them. The principle of virtual work is used to calculate the stresses in the particles and necks. A number of different deformation mechanisms are investigated and, depending upon the deformation mechanism which is dominant for given load conditions, different equations are used to calculate the strains in the grains and necks. These strains around a representative ice grain are then averaged and scaled to obtain the global strains in the snow. The theory is then compared with experimental data to determine if the mechanical properties of snow can be adequately represented. Results show that the constitutive theory does work, but that it is cumbersome to implement, and that for practical use substantial computational capability is needed.